Full-range sharp local maximal bounds in higher dimensions or for matrix weights

Establish the quantitative bound \eqref{eq-bound-maxloc-1} with its sharp exponent for the local maximal operator M_r throughout the full range p\in(1,\infty) when either the dimension n exceeds one or the matrix size m exceeds one.

Background

Theorem 4.1 proves the sharp local maximal-operator estimate in the one-dimensional scalar setting, namely n=m=1, for every p\in(1,\infty). The authors explain that the proof uses separation of weights and operators, interpolation, and the total ordering of the real line.

Those structural features are unavailable in higher-dimensional or matrix-valued settings. The authors therefore explicitly identify the extension of the sharp estimate to these settings as unresolved, even for the ordinary local maximal operator M_r.

References

which, even for $M_r$, is still unknown when $p\in (2,\infty)$ and $n>1$ or $m>1$.

— Local Matrix Muckenhoupt Weights and Quantitative Weighted Inequalities Achieving Global Best Known Exponents  (2609.19597 - Hytönen et al., 17 Sep 2026) in Remark following Theorem 4.1, Section 4

Regarding the lower boundary of ${\mathfrak{W}E$ one may also ask the question when the condition $\sup{j\ge k\ge 0} \Lambda_{k,j}E(p,\alpha)<\infty $ is enough for $Lp(w_\alpha)$ boundedness when $\alpha= L(p)$. This is is mostly unknown; indeed we shall see that there are instances when this condition holds with $\alpha=L(p)$ but $Lp(w_\alpha)$ boundedness fails (see Theorem \ref{thm:sequences} below for the case $\beta<1$, $p=\frac{d}{d-1}$ and $\alpha=1-d$).

— Endpoint power weight inequalities for spherical maximal functions  (2610.18316 - Fraccaroli et al., 13 Oct 2026) in Introduction, subsection 'Two necessary conditions'